A225473 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(-21 - 36\,x - 4\,x^{2} + 16\,x^{3}\right)\,P_{n-2}(x) + \left(-12\,x - 12\,x^{2} + 16\,x^{3} + 16\,x^{4}\right)\,P_{n-2}'(x) + \left(10 + 12\,x - 4\,x^{2}\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 7 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	4														11
3	49	104	32													185
4	343	1660	1824	384												4211
5	2401	22160	51392	38400	6144											120497
6	16807	271924	1106880	1678080	952320	122880										4148891
7	117649	3190904	20821472	52999680	59566080	27279360	2949120									166924265
8	823543	36505900	362441184	1411127424	2590586880	2311249920	887685120	82575360								7682995331
9	5764801	411429920	6014141312	33917506560	92531625984	132347658240	97851801600	32369541120	2642411520							398091881057
10	40353607	4594854244	96748932480	763092764160	2926750740480	6100311490560	7133272473600	4501678325760	1307993702400	95126814720						22929610452011
11	282475249	51027482504	1524428983712	16415523087360	85377568911360	245194226565120	411951254077440	406954495180800	223984012492800	58027356979200	3805072588800					1453285248824345
12	1977326743	564691538140	23680615146144	342180538068864	2353659155189760	8978984371568640	20486466351267840	28768433138565120	24586357624012800	11996918238412800	2804338497945600	167423193907200				100509008392949651
13	13841287201	6235333423280	364228474605632	6972791668677120	62284095461812224	307666272806830080	918775854445363200	1733041319722352640	2087815972588093440	1571622449106124800	688695308137267200	146662717862707200	8036313307545600			7531943572756089617
14	96889010407	68754758850964	5563081727810880	139742725226307840	1598918215061207040	10037239200529981440	38230458632522956800	93338307517423288320	149752741805162496000	157834843525050531840	106129478906177126400	42202364488187904000	8253293766849331200	417888291992371200		607940909007559174331
15	678223072849	757465002730904	84545526959449952	2766098782370327040	40114201011060234240	315499368820765409280	1503832762804124712960	4630171473150109286400	9532049535622604390400	13287910565421357465600	12440834265697379942400	7557117990408290304000	2750436265820789145600	497287067470921728000	23401744351572787200	52581506643031530986825

Technical details

Node id: n-29f6988ef49acca3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6787cdb08a7f183b

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        7,          4;
   3:       49,        104,          32;
   4:      343,       1660,        1824,          384;
   5:     2401,      22160,       51392,        38400,          6144;
   6:    16807,     271924,     1106880,      1678080,        952320,        122880;
   7:   117649,    3190904,    20821472,     52999680,      59566080,      27279360,       2949120;
   8:   823543,   36505900,   362441184,   1411127424,    2590586880,    2311249920,     887685120,      82575360;
   9:  5764801,  411429920,  6014141312,  33917506560,   92531625984,  132347658240,   97851801600,   32369541120,    2642411520;
  10: 40353607, 4594854244, 96748932480, 763092764160, 2926750740480, 6100311490560, 7133272473600, 4501678325760, 1307993702400, 95126814720;

%F Formula

The row polynomials satisfy P(n,x) = -(21+36*x+4*x^2-16*x^3)*P(n-2,x) - (12*x+12*x^2-16*x^3-16*x^4)*d/dx P(n-2,x) + (10+12*x-4*x^2)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(21 + 36*x + 4*x^2 - 16*x^3)*P[n - 2, x] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*D[P[n - 2, x], x] + (10 + 12*x - 4*x^2)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 4*x; for(n = 3, N, my(i = n); v[i] = -(21 + 36*x + 4*x^2 - 16*x^3)*v[i - 2] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*deriv(v[i - 2], x) + (10 + 12*x - 4*x^2)*v[i - 1] + (4*x + 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.